Konferensi Nasional Teknik Sipil 3 (KoNTekS 3) Jakarta, 6 7 Mei 2009 ONE-DIMENSIONAL CONSOLIDATION THROUGH FLUID-SATURATED NONLINEAR POROUS MEDIA

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1 onfeens Nasonal Teknk pl (ontek ) Jakata, 6 7 e 9 ON-DINIONAL CONOLIDATION THROUGH LUID-ATURATD NONLINAR POROU DIA Jack Wdjajakusuma Juusan Teknk pl, Unvestas Pelta Haapan e-mal: jack_w@uphedu ABTRACT In ths pape we apply the theoy of poous meda n studyng consoldaton though flud satuated poous meda The theoy of poous meda s defned as the theoy of mxtues estcted by the volume facton concept nce the sold matx can have fnte elastc defomaton, we poposed a nonlnea model to study such consoldaton though nonlnea poous meda It s to fnd out that the nonlnea model gves the satsfactoy esult and the theoy of poous meda can captue the flud stuctue nteacton phenomena eywods: theoy of poous meda, consoldaton, non-lnea elastcty, mxtue theoy, pemeablty 1 INTRODUCTION Poous meda can be consdeed as a two-phase medum, whose phases fom nfnte paths acoss the heteogeneous andom Consoldaton pocess though poous meda occu n many cvl engneeng poblems o example, desgn of foundatons of dams o buldngs, excavatons and settlement poblems The consoldaton pocess of the nonlnea poous meda cannot be studed usng nethe the method of sold mechancs and no the method of flud mechancs alone Instead, t has to be taken nto consdeaton the couplng effect between poe flud and sold matx One way to coectly teat couplng effect s usng the theoy of poous meda (Bowen, 198, de Boe & hles, 1986, hles, 1989, hles, 1996, de Boe, ) The theoy of poous meda s defned as the theoy of mxtues (theoy of heteogeneous contnua wth ntenal nteactons) extended by the volume facton concept The pesent pape studes consoldaton pocess n flud-satuated nonlnea poous solds usng the theoy of poous meda In ths case, the poous medum s modeled as a flud-satuated bphasc medum, whch s composed of a flud phase ϕ and a sold phase ϕ The poe flud s assumed to be an ncompessble vscous flud and the sold matx s modeled as the ncompessble Ogden mateal (Ogden, 197, Ogden, 1986, Ogden, 1) Ths assumpton allows that the sold matx to have fnte elastc defomaton The outlne of the pape s as follows ecton summazes the basc feld equatons, whch ae needed to analyze consoldaton phenomena n nonlnea flud-satuated poous medum ecton deals wth the dscetzaton of the govenng feld equatons ecton 4 pesents some numecal examples to llustate the method ecton 5 concludes the pape BAIC ILD QUATION Ths secton summazes the basc feld equatons, whch ae needed n studyng consoldaton phenomena n poous medum These equatons ae deved n the famewok of the theoy of poous meda The theoy of poous meda s defned as the theoy of heteogeneous contnua wth ntenal nteactons extended by the volume facton concept The detaled dscusson about the theoy of poous meda can be found n de Boe (), hles (1989, 199, 1996) We consde that the phases ϕ and ϕ have the volume V ( ϕ ) and V ( ϕ ) the heteogeneous medum s gven by ( V V ϕ ) V ( ϕ ) defned as ( ϕ ) = + The volume facton of the phases ( ), espectvely The total volume of V V ϕ n = and n =, (1) V V espectvely The flud-satuated poous medum eques ϕ and ϕ ae Unvestas Pelta Haapan Unvestas Atma Jaya Yogyakata G - 7

2 Jack Wdjajakusuma n + n = 1 () In the famewok of the theoy of poous meda, each pont at any tme t each spatal pont x of the actual confguaton s smultaneously occuped by mateal patcles x of consttuents ϕ These mateal patcles poceed fom dffeent efeence postons X at tme t In the case of couplng sold-flud poblems, t s useful to descbe the sold phase vecto u as the pmay knematc vaable, ϕ usng the sold dsplacement u = x X () Then, the velocty v s gven by ( x) v = (4) Next, we defne the seepage velocty w as the elatve moton of the flud to the defomng sold phase Thus, w ( x) ( ) x = (5) Theen, the mateal tme devatve of any feld quantty Γ efeed to a movng patcle of the phase s gven by ( ) Γ Γ ( )( ) = + gad Γ x, (6) t whee =, and the opeato gad means the patal dffeentaton wth espect to x ϕ, ( Γ ), In ths pape, we consde that thee s no mass and heat exchanges between the sold and flud phase In ths case, the volume balance of the poous medum eads (de Boe,, hles, 1989, 199, Wdjajakusuma 8) ( n w v ) = dv (7) + The balance of the lnea momentum fo flud can be wtten as (de Boe,, hles, 1989, 199, Wdjajakusuma 8) whee ρ {( v + w ) + [ ( v + w )] w } = dvt + ρ f + pˆ ρ s the patal densty of flud, gad, (8) T s the patal Cauchy stess tenso fo flud, f s the body foce densty actng upon both consttuents The balance of the lnea momentum fo mxtue s gven by (de Boe,, hles, 1989, 199, Wdjajakusuma, 8) whee {( ) + gad ( ) } ( )( ) dv = ( + ) + ( + ) ρ w v w w ρ ρ v T T ρ ρ f, (9) ρ s the patal densty of the sold matx, The patal Cauchy stess tenso T s the patal Cauchy stess tenso fo sold T can be detemned by sepaated nto two pats The fst pat esults fom the poe-flud pessue and the second pat, the exta tem ( ), esults fom the sold defomaton (effectve stess) and the poe-flud flow (fctonal stess) Ths sepaaton s known as the effectve stess pncple (Das, ) Thus, the flud and the sold patal stess tensos ae gven by T = n pi + T, (1) and T = n pi + T, (11) espectvely (hles 1989, 199) Hee, I s the second ode dentty tenso In ode to captue the nonlnea effect of poous medum, we model the sold phase ϕ as an Ogden s mateal (Ogden, 197, Ogden, 1986, Ogden, 1) Thus, the sold patal stess tenso s gven by G - 74 Unvestas Pelta Haapan Unvestas Atma Jaya Yogyakata

3 One-Dmensonal Condoldaton Though lud-atuated Nonlnea Poous eda whee Hee, µ, T 1 ( ) τ = det, (1) τ s the chhoff stess tenso and s gven by (Wdjajakusuma, 8) τ W = ( λ, λ, λ ) = W 1 B = B B = 1 λ B Λ / β µ µ λ N µ µ I + ( 1 J )I (1) = 1 = 1 = 1 β J = λ1 λ λ = det B, µ,, Λ and β ae Ogden s mateal paamete, egenvalue of the left Cauchy-Geen tenso followng condtons µ λ B uthemoe, the mateal paametes µ, > and µ = (14) = 1 o smplcty, the exta flud stesses ae neglected The nteacton foce s gven by T (15) ˆ p = p n + ˆp λ s the µ, should satfy gad, (16) whee the exta nteacton foce ˆp s the ntenal fcton between sold and flud ˆp may be gven as ( n ) γ ˆp = k R Theen, γ s the effectve weght of the flud and paamete k s sotopc and constant R w (17) k s Dacy pemeablty We assume that the pemeablty DICRTIZATION In ths wok, the bounday value poblems nvolvng mechancs of flud satuated elastome foams ae solved usng Galekn fnte element method (G) (Debels & hles, 1996, Zenkewcs and Taylo,, hles & llsepen, 1) To deve the G method, the govenng feld equatons (4), (7), (8), and (9) ae multpled by ndependent test functons of the sold dsplacement δ u, the flud pessue δ p, the sold velocty δ v and the seepage velocty δ w, espectvely Then, the esultng equatons ae ntegated ove the spatal doman V wth bounday V and by applyng Geen s theoem (ntegaton by pats), the ode of the spatal devatves can be educed and the bounday condtons can be appled o detaled see Debels & hles, 1996, Wdjajakusuma, 8 Afte dscetzaton the system of nonlnea patal dffeental equatons educes to a system of dffeental-algebac equatons (Benan et al, 1989), whch can be wtten n matx fom as 11 u v + j w j p 1 y 1 u 1 4 v f = j 4 w f 4 4 p j f { The system of equatons (18) may be ntegated by the mplct ule scheme (Benan et al, 1989): y f (18) ( n+ 1) ( + ) ( y y ) + y 1 n ( n 1) ( n+ 1) ( n) + 1 = t f (19) Unvestas Pelta Haapan Unvestas Atma Jaya Yogyakata G - 75

4 Jack Wdjajakusuma Theen, t +1 step nally, the non-lnea system of equatons (19) s solved usng Newton teaton denotes the tme ncement, ( n ) ndcates the tme step to be computed, ( ) n s the cuent tme 4 ON-DINIONAL OIL COLUN In ode to llustate the consoldaton phenomena of nonlnea poous medum, we use a fully satuated sol column of 1 m depth We choose the uppe bounday s pefectly daned and subjected unde dynamcal suface loadng, meanwhle the othe boundaes ae gd and undaned (see gue 1) gue 1 ol column unde extenal load The mateal paametes fo lnea poous medum ae taken fom de Boe et al, 199: µ = and λ = 8, Lamé constants: N m ffectve denstes: ρ R = kg m and 7 effectve tue weght of flud γ R = 1 N m the efeence volume factons: 67 eanwhle, the Ogden mateal paametes ae chosen as ρ R 6 N = 1 kg n = and n = µ () 1 = , µ ( ) = 8 and µ ( ) = 7 () = 1, ( ) = 5 and ( ) = 1 Due to the assumpton of ncompessble consttuents, the lagest possble vetcal dsplacement possble of ths example s - m (theoetcal lmt), whch coesponds to compessve volume stans of % In ths state, all of the poe flud daned out and all poes ae closed, thus, the sol column locks and no futhe defomaton s possble 6 It s seen fom gue that the unde the load P = 15 1 N m, the nonlnea model ( β = 9 ) gves solutons n the admssble doman All of the dsplacements of ths nonlnea model do not exceed the theoetcal lmt, meanwhle the dsplacement of lnea model exceeds the theoetcal lmt unde the same bounday condtons (see also Debels & hles, 1996, Wdjajakusuma, 8, 9) The poe wate pessue foces the wate to dan out of the poe space As wate stats escapng fom the poe space, the poe wate pessue n wate gets gadually dsspated and the pessue ncement s shfted As the wate escapes out of system, the load tansfe takes place fom wate to the sold Thus, the dsplacement of the sol column wth the lagest pemeablty wll fstly each the theoetcal lmt of - m and the poe wate pessue of m m G - 76 Unvestas Pelta Haapan Unvestas Atma Jaya Yogyakata

5 One-Dmensonal Condoldaton Though lud-atuated Nonlnea Poous eda ths sol column s the lowest (gue ) Theefoe, the smalle value of Dacy pemeablty s, the longe consoldaton pocess wll be nce the wate can solely dan out fom the uppe bounday, the poe wate pessue of uppe pat s almost zeo and gadually ncease wth the nceasng depth of sol column (gue b) gue (a) Vetcal dsplacement of sol column wth dffeent Dacy pemeablty (b) Poe wate pessue at t = s 5 CONCLUDING RAR In ths pape, a model, whch s based on the theoy of poous meda and can descbe one-dmensonal consoldaton pocess of nonlnea poous medum, has been gven In the pesent model, the sold-matx and the poe-flud ae assumed to be ncompessble Thus, the dsplacement of the sold matx esults fom the vaaton of the poe space, and the maxmum of the asng volume stans equals the maxmum of the poosty n The statng pont fo the fnte element analyss ae the weak fomulaton of the elaton between the dsplacement and the velocty, of the the volume balance equaton of the mxtue, of the balance of the lnea momentum fo mxtue and of the equaton of moton of the poe flud It s to fnd out that ou nonlnea model gves the admssble fnte dsplacements unde the gven ntal bounday-value poblem RRNC Benan,, Campbell, and Petzold, L (1989) Numecal olutons of Intal-Value Poblems n Dffeental- Algebac quatons Noth-Holland, Amstedam Das, B () Pncples of Geotechncal ngneeng Books/Cole, Calfona, UA de Boe, R () Theoy of Poous eda, pnge-velag, Beln de Boe, R, hles, W and Z Lu (199) One-dmensonal tansent wave popagaton n flud satuated ncompessble poous meda Ach Appl ech 6, 59-7 de Boe, R and hles, W (1986) Theoe de ehkomponentenkontnua mt Anwendung auf bodenmechansche Pobleme, Tel I, oschungsbechte aus dem achbeech Bauwesen 4, Unvestät-Gesamthochschule ssen, ssen Debels, and hles, W (1996) Dynamc analyss of a fully satuated poous medum accountng fo geometcal and mateal non-lneates Int J Nume eth ng 9, hles, W (1989) Poöse eden en kontnuumsmechansches odell auf de Bass de schungstheoe, oschungsbechte aus dem achbeech Bauwesen 47, Unvestät-Gesamthochschule ssen, ssen hles, W (199) Consttutve equatons fo ganula mateals n geometcal context, n Hutte (ed), Contnuum echancs n nvonmental cences and Geophyscs pnge Velag, Wen, 1-4 hles, W (1996) Gundlegende onzepte n de Theoe Poöse eden Technsche echank 16, 6-76 Unvestas Pelta Haapan Unvestas Atma Jaya Yogyakata G - 77

6 Jack Wdjajakusuma hles, W and llsepen, P (1) Theoetcal and Numecal ethods n nvonmental Contnuum echancs Based on the Theoy of Poous eda, n chefle, B (ed), nvonmental Geomechancs, CI Couses and Lectues 417 pnge-velag, Wen, pp 1-81 Ogden, R W (197) Lage Defomaton Isotopc lastcty: on the Coelaton of Theoy and xpement fo Compessble Rubbelke olds Poc Royal oc Lond A Vol 8, Ogden, R W (1986) Recent Advances n the Phenomenologcal Theoy of Rubbe lastcty Rubbe Chemsty and Technology, Vol 59, 61-8 Ogden, R W (1) lements of the Theoy of nte lastcty, n Y B u and R W Ogden (eds), Nonlnea lastcty, Theoy and Applcatons Cambdge Unvesty Pess, Cambdge, pp1-57 Wdjajakusuma, J (8) Behavou of lastome oams, n Posdng onfeens Nasonal Teknk pl, Unvestas Atma Jaya, Yogyakata, Wdjajakusuma, J (9) Behavou of Nonlnea Poous eda, to be publshed Zenkewcz, O C (1984) Coupled poblems and the numecal soluton, In R W Lews, P Bettess and Hnton (eds), Numecal ethods n Coupled ystems John Wley and ons, Chcheste, pp Zenkewcz, O C and Taylo, R L (), The nte lement ethod Volume 1: The Bass, Buttewoth- Henemann, Oxfod G - 78 Unvestas Pelta Haapan Unvestas Atma Jaya Yogyakata